If now b be moved parallel with itself a measured distance by means of the micrometer screw, the number of alternations of light and darkness is exactly twice the number of wave lengths in the measured distance.  Thus the determination consists absolutely of a measurement of a length and the counting of a number.

The degree of accuracy depends on the number of wave lengths which it is possible to count.  Fizeau was unable to observe interference when the difference of path amounted to 50,000 wave lengths.  It seemed probable that with a smaller density of sodium vapor this number might be increased, and the experiment was tried with metallic sodium in an exhausted tube provided with aluminum electrodes.  It was found possible to increase this number to more than 200,000.  Now it is very easy to estimate tenths or even twentieths of a wave length, which implies that it is possible to find the number of wave lengths in a given fixed distance between two planes with an error less than one part in two millions and probably one in ten millions.  But the distance corresponding to 400,000 wave lengths is roughly a decimeter, and this cannot be determined or reproduced more accurately than say to one part in 500,000.  So it would be necessary to increase this distance.  This can be done by using the same instrument together with a comparer.

The intermediate standard decimeter, lm (Fig. 2), is put in place of the mirror, b.  It consists of a prism of glass one decimeter long with one end, l, plane, and the other slightly convex, so that when it touches the plane, m, Newton’s rings appear, and these serve to control any change in the distance, lm, which has been previously determined in wave lengths.

The end, l, is now adjusted so that colored fringes appear in white light.  These can be measured to within one-twentieth of a wave length, and probably to within one-fiftieth.  The piece, lm, is then moved forward till the fringes again appear at m.  Then the refractometer is moved in the same direction till the fringes appear again at l, and so on till the whole meter has been stepped off.  Supposing that in this operation the error in the setting of the fringes is always in the same direction, the whole error in stepping off the meter would be one part in two millions.  By repetition this could of course be reduced.  A microscope rigidly attached to the carriage holding the piece, lm, would serve to compare, and a diamond attached to the same piece would be used to produce copies.  All measurements would be made with the apparatus surrounded by melting ice, so that no temperature corrections would be required.

Probably there would be considerable difficulty in actually counting 400,000 wave lengths, but this can be avoided by first counting the wave lengths and fractions in a length of one millimeter and using this to step off a centimeter.  This will give the nearest whole number of wave lengths, and the fractions may be observed directly.  The centimeter is then used in the same way to step off a decimeter, which again determines the nearest whole number, the fraction being observed directly as before.